Numerical Results

The classical version of the problem was treated both numerically with Monte Carlo simulations by Ioannis Rousochatzakis et al. [7], and analytically with Luttinger-Tisza approximation by Michael Becker at al. [6].

The quantum problem was treated numerically using exact diagonalization on small clusters of up to 27 sites [6]. The resulting phase diagram is shown in Figure 2.10. In this work M. Becker at al. also examined AF Kitaev point using the density-matrix renormalization group (DMRG) on small clusters. Kazuya Shinjo et al. [18] used the density-matrix renormalization group (DMRG) on lattices with 12

× 6 sites to study the Heisenberg-Kitaev model.

Exact diagonalization on a 12-site cluster and a Schwinger-fermion mean-field method for the point J_H = 0, J_K >0 was used by Kai Li et al. [19].

Numerical results point towards the existence of 5 different phases mentioned below.

Z2 Vortex phase

Classical numerical treatments of the problem suggests that Kitaev coupling close to antiferromagnetic Heisenberg point ψ = 0 changes the 120-degree order to an incommensurate non-coplanarZ²vortex phase. In the long distance limit aZ²vortex phase can be understood as a 120-degree order with a slowly varying coordinate

Figure 2.11: Left: Bragg peaks ofS^γγ(q) are shifted. Blue, red and black correspond to γ=x,y and z respectively. Right: One of the three sublattices of the Z² vortex crystal. Both figures correspond to the classical model and are taken from the paper of M. Becker et al. [6]

frame. The Bragg peaks in the static spin structure factors² S^γγ(q) are shifted from the corners of BZ to incommensurate momenta in γ direction, as is indicated in Figure 2.11.

What happens in the quantum model is not enteirly clear, DMRG [18] suggests that S^γγ(q) are not just delta functions, and that S^γγ(q) are different for different γ.

Nematic Phase

The exact diagonalization data and DMRG point toward the extance of a nematic phase in the vicinity of the Kitaev AF pointψ = ^π₂. Classically, the phase atJ_H = 0, J_K > 0 has a large ground state degeneracy consisting of antiferromagnetic Ising chains that are decoupled [7]. Actually, these are not all possible classical ground states, the whole ground state manifold has massive SO(2) degeneracy and can be written as [7]:

S_r= (f_xxn−m(−1)^m, f_yy_m(−1)ⁿ, f_zz_n(−1)^m)^T, (2.12) with f_x²+f_y² +f_z² = 1. The lattice coordinates are (n, m) = na_x +ma_y and the sets{x_m},{y_m}and{z_m}are random choices of±1. These states contain collinear, coplanar and non-coplanar states.

DMRG suggest that this degeneracy is reduced to non-extensive 3×2² [6, 18]

and that second nearest AF Ising chains are aligned. This was also shown ana-lytically with quantum order-by-disorder method [20]. Numerical results suggest that for small J_H > 0, neighbouring spin chains are aligned antiferromagnetically.

Correlations between spins are shown in Figure 2.12.

Z⁶ Ferromagnetic Phase

At the FM Heisenberg point J_K = 0, J_H < 0 the model is SU(2) symmetric, and the ground state consists of spins aligned in the same direction. At this point the ferromagnetic order parameter can point in any direction.

2Defined in section 4 with the equation 4.14.

Figure 2.12: < S_i^xS_j^x > for the nematic phase at the Kitaev AF point, the one of the 6 degenerate states that is expected for JH > 0. Upward red (downward blue) arrows correspond to positive (negative) correlations. Figure from [18].

For finite JK, this degeneracy is reduced toZ⁶ by quantum fluctuations, i.e. six directions along the spin axes. This has been shown by quantum order by disorder [7], exact diagonalization and by 1/S term of large S expansion [6].

Effects of finite symmetric anisotropic term Γ.

Andrei Catuneanu et al. [15] tried to explain the physics of Ba₃IrTi₂O₉. The theoretical analysis showed that an additional symmetric anisotropic term should be included in the effective Hamiltonian in addition to the Heisenberg-Kitaev part.

It can be written as:

HΓ= X

αβ⊥hi,ji

Γ(S_i^αS_j^β+S_i^βS_j^α), (2.13) where the sum runs over nearest neighbours. αβ ⊥ hi, ji means that on γ bond we sum over indices that are different from γ, i.e. on x bond we get a term Γ(S_i^yS_j^z+ S_i^zS_j^y).

They used Luttinger-Tisza method and classical Monte Carlo simulations to show that a finite value of Γ stabilizes the 120-degree order and stripy phase in a large part of the phase diagram. Specifically, they predict that Ba₃IrTi₂O₉ has a stripy ordered ground state.

Chapter 3

Schwinger-boson Mean-field Theory (SBMFT)

Obtaining ground states of the spin Hamiltonians on the triangular lattice is no-toriously difficult. Schwinger-boson mean-field theory (SBMFT) provides a way of obtaining an approximate solution to this problem. This is done by representing spin operators with bosons, choosing a field ansatz, preforming the mean-field decoupling and diagonalizing the quadratic Hamiltonian using a Bogoliubov transformation. Mean-field parameters need to be determined self-consistently. The mean-field decoupling has been shown to be equivalent to the large N limit of the symplectic group Sp(N), which is a generalization of SU(2) spin algebra [21].

3.1 Schwinger-bosons Representation

We can express the spin operators with two species b_↑ and b_↓ of Schwinger-bosons:

Si = 1

2b^†_iασαβbiβ, (3.1)

where the indices α and β run over up and down values (we employ a summation convention over the repeated Greek indices), i denotes the lattice site and σ_αβ is a vector of Pauli matrices. The commutation relations are preserved, but the resulting Hilbert space is too big. To reobtain the original problem, we need to restrict ourselves to the case where the density of bosons is equal to 2S

ˆ

n_i =b^†_iαb_iα = 2S. (3.2)

This is one of the advantages of SBMFT, namely we can formally treat the size of spin as a continuous variable. In the large spin limit S → ∞ we arrive at the classical limit, where we expect a long range ordered phase. By decreasing the spin size, the quantum fluctuations become more and more important. For S small enough (formally it can be smaller than 1/2), the spin liquid phase solutions are obtained. We impose the above constraint by adding a Lagrange multiplier term P

iλ_i b^†_iαb_iα−2S

to the Hamiltonian.

The description has a U(1) gauge redundancy:

b_rα →e^iφ(r)b_rα, (3.3a)

b^†_rα →e^−iφ(r)b^†_rα. (3.3b)